Group #14518884 — subgroups

Subgroups — C₃²×3₋¹⁺²⋊Q₈ #14518884

Order 1944.

810 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 270 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₃²×3₋¹⁺²⋊C₄	C₂	1	2	yes	maximal
3	C₃³×Dic₆	—	3	1	no	maximal
4	C₃×3₋¹⁺²⋊Q₈	C₃	1	12	yes	maximal, direct product factor
5	C₃²×Dic₁₈	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₃⁴×C₄	D₃	1	1	yes	characteristic
8	C₃³×Dic₃	—	3	2	no	—
9	C₃×C₄×3₋¹⁺²	—	2	12	no	—
10	C₃×C₄×3₋¹⁺²	C₂×C₃	1	12	yes	—
11	C₃²×C₄×C₉	—	2	1	no	—
12	C₃²×C₄×C₉	C₂×C₃	1	1	yes	characteristic
13	C₃×3₋¹⁺²⋊C₄	C₂×C₃	1	24	yes	—
14	C₃²×Dic₉	C₂×C₃	1	2	yes	—
15	C₃²×3₋¹⁺²	Q₈	1	1	yes	characteristic, complemented, Sylow
16	C₃³×Q₈	—	9	1	no	—
17	C₃²×Dic₆	—	3	1	no	—
18	C₃²×Dic₆	—	3	12	no	—
19	3₋¹⁺²⋊Q₈	C₃²	1	9	yes	direct product factor
20	C₃×Dic₁₈	C₃²	1	4	yes	complemented
21	C₂×C₃⁴	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃×3₋¹⁺²	—	2	12	no	—
23	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	12	yes	—
24	C₂×C₃²×C₉	—	2	1	no	—
25	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
26	C₃³×C₄	—	3	1	no	—
27	C₃³×C₄	—	6	4	no	—
28	C₃³×C₄	—	9	2	no	—
29	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
30	C₃³×C₄	C₃×D₃	1	12	yes	—
31	C₃²×Dic₃	—	3	2	no	—
32	C₃²×Dic₃	—	3	24	no	—
33	C₄×3₋¹⁺²	—	2	36	no	—
34	C₄×3₋¹⁺²	C₂×C₃²	1	9	yes	—
35	C₃×C₄×C₉	—	2	12	no	—
36	C₃×C₄×C₉	—	2	4	no	—
37	C₃×C₄×C₉	C₂×C₃²	1	4	yes	—
38	3₋¹⁺²⋊C₄	C₂×C₃²	1	18	yes	—
39	C₃×Dic₉	C₂×C₃²	1	8	yes	—
40	C₃⁴	Dic₆	1	1	yes	characteristic
41	C₃×3₋¹⁺²	—	2	12	no	—
42	C₃×3₋¹⁺²	C₃×Q₈	1	12	yes	complemented
43	C₃²×C₉	—	2	1	no	—
44	C₃²×C₉	C₃×Q₈	1	1	yes	characteristic, complemented
45	C₃²×Q₈	—	9	12	no	—
46	C₃²×Q₈	—	9	1	no	—
47	C₃×Dic₆	—	3	4	no	—
48	C₃×Dic₆	—	3	9	no	—
49	Dic₁₈	C₃³	1	1	yes	characteristic, complemented
50	C₂×C₃³	—	3	1	no	—
51	C₂×C₃³	—	6	4	no	—
52	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
53	C₂×C₃³	C₂×C₃×D₃	1	12	yes	—
54	C₂×3₋¹⁺²	—	2	36	no	—
55	C₂×3₋¹⁺²	C₂²×C₃²	1	9	yes	—
56	C₂×C₃×C₉	—	2	12	no	—
57	C₂×C₃×C₉	—	2	4	no	—
58	C₂×C₃×C₉	C₂²×C₃²	1	4	yes	—
59	C₃²×C₄	—	2	4	no	—
60	C₃²×C₄	—	3	12	no	—
61	C₃²×C₄	—	6	12	no	—
62	C₃²×C₄	—	9	24	no	—
63	C₃²×C₄	—	9	2	no	—
64	C₃²×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic
65	C₃²×C₄	C₃²×D₃	1	4	yes	—
66	C₃²×C₄	C₃²×D₃	1	9	yes	—
67	C₃×Dic₃	—	3	8	no	—
68	C₃×Dic₃	—	3	18	no	—
69	C₄×C₉	—	2	9	no	—
70	C₄×C₉	—	2	4	no	—
71	C₄×C₉	C₂×C₃³	1	1	yes	characteristic
72	Dic₉	C₂×C₃³	1	2	yes	—
73	C₃³	—	3	1	no	—
74	C₃³	—	6	4	no	—
75	C₃³	C₃×Dic₆	1	1	yes	characteristic
76	C₃³	C₃×Dic₆	1	12	yes	—
77	3₋¹⁺²	—	2	36	no	—
78	3₋¹⁺²	C₃²×Q₈	1	9	yes	complemented
79	C₃×C₉	—	2	12	no	—
80	C₃×C₉	—	2	4	no	—
81	C₃×C₉	C₃²×Q₈	1	4	yes	complemented
82	C₃×Q₈	—	9	9	no	—
83	C₃×Q₈	—	9	4	no	—
84	Dic₆	—	3	1	no	—
85	C₂×C₃²	—	2	4	no	—
86	C₂×C₃²	—	3	12	no	—
87	C₂×C₃²	—	6	12	no	—
88	C₂×C₃²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
89	C₂×C₃²	C₂×C₃²×D₃	1	4	yes	—
90	C₂×C₃²	C₂×C₃²×D₃	1	9	yes	—
91	C₂×C₉	—	2	9	no	—
92	C₂×C₉	—	2	4	no	—
93	C₂×C₉	C₂²×C₃³	1	1	yes	characteristic
94	C₃×C₄	—	2	4	no	—
95	C₃×C₄	—	3	9	no	—
96	C₃×C₄	—	9	18	no	—
97	C₃×C₄	—	9	8	no	—
98	C₃×C₄	C₃×AGL(1,ℤ/9)	1	4	yes	—
99	C₃×C₄	C₃³×D₃	1	1	yes	characteristic
100	Dic₃	—	3	2	no	—
101	C₃²	—	2	4	no	—
102	C₃²	—	3	12	no	—
103	C₃²	—	6	12	no	—
104	C₃²	3₋¹⁺²⋊Q₈	1	1	yes	characteristic, direct product factor
105	C₃²	C₃²×Dic₆	1	4	yes	—
106	C₃²	C₃²×Dic₆	1	9	yes	—
107	C₉	—	2	9	no	—
108	C₉	—	2	4	no	—
109	C₉	C₃³×Q₈	1	1	yes	characteristic, complemented
110	Q₈	—	9	1	no	Sylow
111	C₂×C₃	—	2	4	no	—
112	C₂×C₃	—	3	9	no	—
113	C₂×C₃	C₂×C₃×AGL(1,ℤ/9)	1	4	yes	—
114	C₂×C₃	C₂×C₃³×D₃	1	1	yes	characteristic
115	C₄	—	9	2	no	—
116	C₄	C₃²×AGL(1,ℤ/9)	1	1	yes	characteristic
117	C₃	—	2	4	no	—
118	C₃	—	3	9	no	—
119	C₃	C₃×3₋¹⁺²⋊Q₈	1	4	yes	direct product factor
120	C₃	C₃³×Dic₆	1	1	yes	characteristic
121	C₂	C₂×C₃²×AGL(1,ℤ/9)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₄×3₋¹⁺²

C₂

yes

maximal, characteristic

C₃²×3₋¹⁺²⋊C₄

C₂

yes

maximal

C₃³×Dic₆

—

maximal

C₃×3₋¹⁺²⋊Q₈

C₃

yes

maximal, direct product factor

C₃²×Dic₁₈

C₃

yes

maximal, characteristic, complemented

C₂×C₃²×3₋¹⁺²

C₂²

yes

characteristic

C₃⁴×C₄

D₃

yes

characteristic

C₃³×Dic₃

—

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

—

C₃²×C₄×C₉

—

C₃²×C₄×C₉

C₂×C₃

yes

characteristic

C₃×3₋¹⁺²⋊C₄

C₂×C₃

yes

—

C₃²×Dic₉

C₂×C₃

yes

—

C₃²×3₋¹⁺²

Q₈

yes

characteristic, complemented, Sylow

C₃³×Q₈

—

C₃²×Dic₆

—

C₃²×Dic₆

—

3₋¹⁺²⋊Q₈

C₃²

yes

direct product factor

C₃×Dic₁₈

C₃²

yes

complemented

C₂×C₃⁴

C₂×D₃

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

C₂²×C₃

yes

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

C₂²×C₃

yes

characteristic

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

C₃×D₃

yes

characteristic

C₃³×C₄

C₃×D₃

yes

—

C₃²×Dic₃

—

C₃²×Dic₃

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×C₃²

yes

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×C₃²

yes

—

3₋¹⁺²⋊C₄

C₂×C₃²

yes

—

C₃×Dic₉

C₂×C₃²

yes

—

C₃⁴

Dic₆

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃×Q₈

yes

complemented

C₃²×C₉

—

C₃²×C₉

C₃×Q₈

yes

characteristic, complemented

C₃²×Q₈

—

C₃²×Q₈

—

C₃×Dic₆

—

C₃×Dic₆

—

Dic₁₈

C₃³

yes

characteristic, complemented

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₂×C₃³

C₂×C₃×D₃

yes

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²×C₃²

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

AGL(1,ℤ/9)

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

—

C₃²×C₄

C₃²×D₃

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₃³

yes

characteristic

Dic₉

C₂×C₃³

yes

—

C₃³

—

C₃³

—

C₃³

C₃×Dic₆

yes

characteristic

C₃³

C₃×Dic₆

yes

—

3₋¹⁺²

—

3₋¹⁺²

C₃²×Q₈

yes

complemented

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃²×Q₈

yes

complemented

C₃×Q₈

—

C₃×Q₈

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×AGL(1,ℤ/9)

yes

characteristic

C₂×C₃²

C₂×C₃²×D₃

yes

—

C₂×C₃²

C₂×C₃²×D₃

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×C₃³

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃×AGL(1,ℤ/9)

yes

—

C₃×C₄

C₃³×D₃

yes

characteristic

100

Dic₃

—

101

C₃²

—

102

C₃²

—

103

C₃²

—

104

C₃²

3₋¹⁺²⋊Q₈

yes

characteristic, direct product factor

105

C₃²

C₃²×Dic₆

yes

—

106

C₃²

C₃²×Dic₆

yes

—

107

C₉

—

108

C₉

—

109

C₉

C₃³×Q₈

yes

characteristic, complemented

110

Q₈

—

Sylow

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

C₂×C₃×AGL(1,ℤ/9)

yes

—

114

C₂×C₃

C₂×C₃³×D₃

yes

characteristic

115

C₄

—

116

C₄

C₃²×AGL(1,ℤ/9)

yes

characteristic

117

C₃

—

118

C₃

—

119

C₃

C₃×3₋¹⁺²⋊Q₈

yes

direct product factor

120

C₃

C₃³×Dic₆

yes

characteristic

121

C₂

C₂×C₃²×AGL(1,ℤ/9)

yes

characteristic

	subgroup	G/H
center	C₂×C₃²	C₂×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₉	C₂²×C₃³
Frattini	C₂×C₃	C₂×C₃³×D₃
Fitting	C₃²×C₄×3₋¹⁺²	C₂
socle	C₂×C₃³	C₂×C₃×D₃
prime radical	C₂×C₃⁴	C₂×D₃
hypercenter	C₃²×C₄	AGL(1,ℤ/9)
hypocenter	C₉	C₃³×Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄

subgroup

G/H

center

C₂×C₃²

C₂×AGL(1,ℤ/9)

commutator, cosocle

C₂×C₉

C₂²×C₃³

Frattini

C₂×C₃

C₂×C₃³×D₃

Fitting

C₃²×C₄×3₋¹⁺²

C₂

socle

C₂×C₃³

C₂×C₃×D₃

prime radical

C₂×C₃⁴

C₂×D₃

hypercenter

C₃²×C₄

AGL(1,ℤ/9)

hypocenter

C₉

C₃³×Q₈

LCS

G ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₃² ◁ C₃²×C₄

N	H ≅ G/N	type
4. C₃×3₋¹⁺²⋊Q₈	C₃	direct (N × H)
5. C₃²×Dic₁₈	C₃	semidirect
15. C₃²×3₋¹⁺²	Q₈	semidirect
19. 3₋¹⁺²⋊Q₈	C₃²	direct (N × H)
20. C₃×Dic₁₈	C₃²	semidirect
42. C₃×3₋¹⁺²	C₃×Q₈	semidirect
44. C₃²×C₉	C₃×Q₈	semidirect
49. Dic₁₈	C₃³	semidirect
78. 3₋¹⁺²	C₃²×Q₈	semidirect
81. C₃×C₉	C₃²×Q₈	semidirect
104. C₃²	3₋¹⁺²⋊Q₈	direct (N × H)
109. C₉	C₃³×Q₈	semidirect
119. C₃	C₃×3₋¹⁺²⋊Q₈	direct (N × H)

H ≅ G/N

type

4. C₃×3₋¹⁺²⋊Q₈

C₃

direct (N × H)

5. C₃²×Dic₁₈

C₃

semidirect

15. C₃²×3₋¹⁺²

Q₈

semidirect

19. 3₋¹⁺²⋊Q₈

C₃²

direct (N × H)

20. C₃×Dic₁₈

C₃²

semidirect

42. C₃×3₋¹⁺²

C₃×Q₈

semidirect

44. C₃²×C₉

C₃×Q₈

semidirect

49. Dic₁₈

C₃³

semidirect

78. 3₋¹⁺²

C₃²×Q₈

semidirect

81. C₃×C₉

C₃²×Q₈

semidirect

104. C₃²

3₋¹⁺²⋊Q₈

direct (N × H)

109. C₉

C₃³×Q₈

semidirect

119. C₃

C₃×3₋¹⁺²⋊Q₈

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₄×3₋¹⁺²	2
C₃²×3₋¹⁺²⋊C₄	2
C₃³×Dic₆	3
C₃×3₋¹⁺²⋊Q₈	3
C₃²×Dic₁₈	3

subgroup

index

C₃²×C₄×3₋¹⁺²

C₃²×3₋¹⁺²⋊C₄

C₃³×Dic₆

C₃×3₋¹⁺²⋊Q₈

C₃²×Dic₁₈

Subgroups — C₃²×3₋¹⁺²⋊Q₈ #14518884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C32×3−1+2⋊Q8 #14518884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×3₋¹⁺²⋊Q₈ #14518884